ftp access:

Published in:

Homotopies and automorphisms of crossed modules of groupoids

Abstract:

We give a detailed description of the structure of the actor 2-crossed module
related to the automorphisms of a crossed module of groupoids. This generalises
work of Brown and Gilbert for the case of crossed modules of groups, and
part of this is needed for work on 2-dimensional holonomy to be developed
elsewhere.

ftp access:

Published in:

Towards a 2-dimensional notion of holonomy

Abstract:

Previous work (Pradines, C.R.Acad.Sci.Paris 263 (1966) 907;
Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for
a holonomy Lie groupoid of a locally Lie groupoid.
Here we develop analogous 2-dimensional notions
starting from a locally Lie crossed module of groupoids.
This involves replacing the Ehresmann notion of a local smooth coadmissible
section of a groupoid by a local smooth coadmissible homotopy
(or free derivation) for the crossed module case.
The development also has to use
corresponding notions for certain types of double groupoids.
This leads to a holonomy Lie groupoid rather than double groupoid,
but one which involves the 2-dimensional information.

Published in:

Local subgroupoids II: Examples and properties

Abstract:

The notion of local subgroupoid as a generalisation of a local equivalence
relation was defined in a previous paper by the first two authors. Here we
use the notion of star path connectivity for a Lie groupoid to give an important
new class of examples, generalising the local equivalence relation of a
foliation, and develop in this new context basic properties of coherence,
due earlier to Rosenthal in the special case. These results are required
for further applications to holonomy and monodromy.

ftp access:

Published in:

Sheaves and Local Subgroupoids

Abstract:

This is an introduction to the notion of local subgroupoid introduced
by the author and R. Brown. It can also serve as an introduction to an
application of sheaf theory, and so could be useful to beginners in that
theory.
The main results are the construction of the holonomy groupoid and the notion
of s-sheaf for certain local subgroupoids s.